Trajectory driven multidisciplinary design optimization of

Struct Multidisc Optim (2015) 52:755–771DOI 10.1007/s00158-015-1267-3

RESEARCH PAPER

Trajectory driven multidisciplinary design optimizationof a sub-orbital spaceplane using non-stationaryGaussian process

Robin Dufour1 · Julien de Muelenaere2 ·Ali Elham1

Received: 10 September 2014 / Revised: 8 May 2015 / Accepted: 8 May 2015 / Published online: 11 June 2015© The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract This paper presents the multidisciplinary opti-mization of an aircraft carried sub-orbital spaceplane. Theoptimization process focused on three disciplines: the aero-dynamics, the structure and the trajectory. The optimizationof the spaceplane geometry was coupled with the optimiza-tion of its trajectory. The structural weight was estimatedusing empirical formulas. The trajectory was optimizedusing a pseudo-spectral approach with an automated meshrefinement that allowed for increasing the sparsity of theJacobian of the constraints. The aerodynamics of the space-plane was computed using an Euler code and the resultswere used to create a surrogate model based on a non-stationary Gaussian process procedure that was speciallydeveloped for this study.

Keywords Spaceplane multidisciplinary optimization ·Optimal control · Surrogate modeling · Gaussian processes

1 Introduction

Access to space is one of the greatest challenge in todaysaerospace technologies. The prohibitive costs of currentsystems led to the rise of new approaches. Reusable sys-tems, such as the American Space Shuttle, were designedto reduce the expenses of the access to Earth Orbit for

� Ali [email protected]

1 Delft University of Technology, Delft, Netherlands

2 Stanford University, Stanford, CA, USA

satellite commissioning, science missions and ferry to theInternational Space Station.

In the present study a novel launch system is consid-ered. This system’s mission is represented in Fig. 1. Anair-launched suborbital winged spaceplane is mounted pig-gyback on a conventional airliner. The airliner brings thespaceplane up to its maximum altitude. Once the maximumaltitude for the airliner is reached, the spaceplane is releasedand propelled by a rocket engine up to 80 kilometers at morethan 2 km/s. After all the fuel has been burned, which ischaracterized by the main engine cut-off (MECO) point, thespaceplane enters a weightless ballistic flight phase. Fiveseconds after the MECO the Upper-Stage (US) is ejectedfrom the spaceplane payload bay. The US rocket engine isignited and brings the payload to its targeted orbit which isassumed to be a sun-synchronous orbit at 700 km altitude(SSO700) in this study.

Such a system is intrinsically multidisciplinary as thedesign of the structure, the aerodynamics and the trajectoryare highly coupled sub-problems. Finding the best solutionrequires to find the best compromise between the differentdisciplines. Multidisciplinary Design Optimization (MDO)addresses this issue by automating the design process andapplying the optimization theory (Sobieszczanski-Sobieskiand Haftka 1997) in order to find a better solution in lesstime than conventional methods. Applying MDO in thespace industries can potentially lead to a large increase inprofits. Examples of using MDO for spaceplane design andoptimization can be found in the works of Yokoyama et al.(2007), Tsuchiya et al. (2007), Takeshi and Mori (2004),Rowell et al. (1999).

The MDO techniques were applied to the considered sys-tem. Three disciplines were considered in the optimizationof the system. The trajectory, the aerodynamics and thestructure. In the latter, the weight of the spaceplane was

mailto:[email protected]

756 R. Dufour et al.

Fig. 1 Studied system mission.Courtesy of Swiss SpaceSystems

estimated using an empirical approach based on winged-space and hypersonic vehicles data developed by Harloffand Berkowitz (1988). Despite the low fidelity of thisapproach it encountered success in the similar study ofYokoyama et al. (2007). Higher fidelity analysis methodswere employed to compute the aerodynamics. The cost ofthese computations is not compatible with the large numberof aerodynamic evaluations required for the trajectory opti-mization. In order to make the computational time practicala surrogate model was employed to represent the aerody-namics of the vehicle. The surrogate model was developedbased on Gradient Enhanced Gaussian process methodolo-gies (Rasmussen 2006). A major drawback of using suchtechniques for fitting aerodynamic data over the Mach num-ber range that is required for a space vehicle, is the difficultyof fitting properly both the sharply varying region in thetransonic regime and the wide plateau of the hypersonicregime. The considered spaceplane in this study falls intothis category with Mach number varying from zero to 10.This issue was addressed by Gramacy and Lee (2008) andby Ying and et al (2007) by partitioning the design spaceand by implementing a mapping function for the covariancerespectively. In the present study a new method is proposedusing the non-stationary kernel developed by Paciorek andSchervish (2004), Plagemann et al. (2008) combined with aset of sub-level stationary Gaussian processes based on thegradients of the aerodynamic coefficients. This method shallensure both the continuity of the generated models and lim-its the number of hyper-parameters to be optimized whilebeing able to capture accurately the local and global trendsof the aerodynamic characteristics of the spaceplane.

This paper will first present the optimization problemand describe the architecture chosen to address the MDOproblem. The trajectory optimization procedure will bedescribed later. This will show the necessity of having quickdisciplinary evaluations because of the large number ofrequired calculations. The weight estimation procedure willthen be explained in more details. In a next section the aero-dynamic analysis will be explained in details, for whichan Euler formulation was selected with an adjoint formula-tion for the estimation of the gradients of the aerodynamiccoefficients. These methods had a significant success inthe field of aerospace optimization for which the workof Martins et al. (2004) and Reuther and Jameson (1995)are solid references. This will lead to the description ofthe new non-stationary gradient enhanced Gaussian pro-cess method which will show its successful use in solvingthe optimization problem in the last section where the geo-metrical modifications, optimal control problem results andperformance improvements will be presented.

2 Problem definition

The objective of the MDO problem is to maximize the massof the payload mpayload brought to the SSO700 orbit. Thissection presents the mathematical definition of the prob-lem as well as the assumptions that were use to solve theoptimization problem.

The space system is composed of three distinct elements,the carrier, the spaceplane and the US. Several assumptionswere made on each of these elements. The carrier aircraft, a

Gaussian processes based trajectory driven MDO of a sub-orbital spaceplane 757

conventional airliner available on the market, is fixed. It isassumed to have a maximum carrying capacity mmax,carrier

that imposed an upper limit on the possible spaceplanemass. The spaceplane and the US are the two other partsof the system to be optimized. The upper stage has beendesigned by a different company, so there is no freedom tomodify and optimize it. Therefore the total mass of the USis assumed to be fixed for this study.

Before introducing the formulation of problem the fol-lowing subsection will show how the US was treated in theoptimization and how the complexity of the problem wasreduced by the use a performance map.

2.1 Performance map of the US

The US total mass mUS,tot includes the US dry massmUS,dry , the mass of the payload mpayloadand the mass ofthe US required fuel mUS,f uel :

mUS,tot = mUS,dry + mpayload + mUS,f uel (1)

The US dry mass includes a constant basic mass mUS,0

formed by the engine, the avionic, the reaction control sys-tem, etc... and a fuel dependent mass that accounts for theincrease of tank and structure mass for an increased fuel vol-ume. This dependency, modeled by a linear function with aslope c. The basic US mass depends on the design of the USand is assumed to be constant in this analysis. Therefore theUS dry mass is expressed as follows:

mUS,dry = mUS,0 + cmUS,f uel (2)

After the separation from the spaceplane the US is assumedto completely burn its fuel before it reaches the requiredorbit to release the payload. Therefore the total mass thatwill reach the orbit is the dry mass of the US plus the pay-load mass. This mass is called the mass injected into orbitminjected :

minjected = mUS,dry + mpayload (3)

Consequently the payload mass can be expressed as:

mpayload = minjected (1 + c) − cmUS,tot − mUS,0(1 + c) (4)

As the total US mass mUS,tot is assumed to be fixed, max-imizing the payload mass is equivalent to maximize to themass injected into orbit. This assumption is used to decouplethe optimization of the US segment from the optimizationof the spaceplane. Indeed the trajectory of the US solelydepends on the conditions at the separation point, namelythe flight path angle (γf ), the speed (vf ) and the altitude(hf ). Consequently it is possible to optimize independentlythe trajectory of the US according to the separation condi-tions. The optimization of the spaceplane can be performed

separately because the final flight conditions of the space-plane correspond to an optimal, pre-calculated, trajectoryfor the US leading to a given injected mass.

A large number of US trajectory optimization with dif-ferent initial conditions, representing the flight conditionsat the separation point, were pre-computed to obtain a 3Dperformance map. In these optimizations the mass injectedinto orbit is maximized according to the optimal controlvector. An example of these maps is shown in Fig. 2. Thisapproach allows for decoupling the trajectory optimizationof the US from the optimization of the spaceplane. Theinjected mass into orbit being interpolated using the finalflight conditions of the spaceplane. This reduces the size ofthe MDO problem, that makes it faster to be solved withbetter convergence properties.

2.2 Spaceplane constraints and assumptions

Several assumption are made concerning the spaceplane.It is assumed to be a rigid body where only longitudinaldynamics are considered. It means that the roll rate, the rollangle and the sideslip angle are always zero. It is worthwhileto note that the yaw corresponds to the heading. The controlsurfaces were considered to be always at their zero position.The vehicle is controlled only with the use of thrust vector-ing. During the initial glide phase the control surfaces arestill assumed to be at their zero position despite the fact thatthe vehicle is unpowered. The errors associated to this initialphase actuation are negligible.

The spaceplane is also subject to several constraints. Itsmass is limited by the capabilities of the carrier which canlift up to mmax,carrier . Right after the spaceplane is separated

Fig. 2 US performance map depending on US segment initial velocityvf vs. US segment initial flight path angle γf at an altitude of 80 km.The contours represent the ratio M∗

injected of the total injected masswith the baseline total injected mass


from the carrier it must glide to a safe distance before ignit-ing its engine. This glide phase is assumed to last from t = 0

to the time of the main engine ignition tMEI equal to fiveseconds. During this phase, the glide ratio of the spaceplanemust be high enough to perform the separation maneuver.This poses the constraint that L/D > 4 during the initialglide.

During the reentry phase of the spaceplane, after theUS ejection, the spaceplane must be able to be naturallytrimmed and statically stable at angles of attack ensuring ahigh drag. This drag is necessary to maximize the energydissipation before the more dense of the atmosphere areencountered to limit the thermal and structural stress. Thisposes the constraint that the pitching moment coefficient Cm

must have a zero value lying between 35 and 45 degreesof angle of attack for Mach numbers between 5 and 10.Also, the slope of the pitching moment coefficient withrespect to the angle of attack must be negative to ensure thelongitudinal static stability.

Lastly the spaceplane is constrained to burn all its fuelwhen the MECO point, defined as the time tf , is reached.

2.3 Mathematical expression of the problem

The goal of the optimization is to maximize the payloadinjected into a SS0700 orbit. This injected mass is calcu-lated by interpolating the US performance map as a functionof the final state of the spaceplane at the separation point:minjected = minjected (rf , vf , γf ). This final state, occurring attf depends on the spaceplane geometrical variables as wellas the optimal control solution for the spaceplane trajectory.

The geometry variables are selected to be the wingspan b, the length of the spaceplane Lref , the doubledelta sweep angle at the nose λnose and the thickness-to-chord ratio t/c of the wing. The design vector is thereforeα, b, Lref , λnose, t/c, where α = [α1, ..., αN ] is the optimalcontrol vector solution where N is the number of collocationpoints.

The state vector is given by the flight parameters[r, φ, θ, v,�], respectively the distance to the earth cen-ter, the longitude, the latitude, the speed, and the heading.The longitude, latitude and heading are taken into accountin order to add the effect of the rotation of the earth ω

as well as Coriolis effect. Because SSO is slightly retro-grade, this impact negatively the performances of the systemwith a magnitude depending on the initial coordinate. ANorth-American launch site was selected for this studycorresponding to 100 m/s loss in velocity as an order ofmagnitude estimate.

The classical equations of motions of a spaceplane (Betts

2010) with the addition of the Coriolis effect terms were usedin the wind frame. The atmosphere was modeled using theUnited State Standard Atmosphere of 1976.

The problem is expressed as:

maxα,b,Lref ,λnose,t/c

minjected (rf , vf , γf ) (5)

subject to:

r = v sin γ

φ = v sin� cos γ

r cos θ

θ = v cos� cos γ

r

v = −g0

(Re

r

)2

sin γ + T cosα − D

m

+ω2r cos θ(sin γ cos θ − cos γ sin θ cos�)

γ = −g0

(Re

r

)2 cos γ

v+ T sinα + L

mv+ v

rcos γ

+2ω sin� cos θ

+ω2r cos θ

v(cos γ cos θ + sin γ sin θ cos�)

� = L sinβ + T sinα sinβ

mv cos γ+ v cos γ sin� sin θ

r cos θ

+2ω(sin θ − cos θ cos� tan γ ) + ω2 r sin θ cos θ sin�

v cos γ

b = 0

Lref = 0

λnose = 0

t/c = 0 (6)

Under the following constraints:

L

D

(α0, ..., αt,MEI

) ≥ 4 (7)

Cm(α = 35) ≥ 0 for M > 4 (8)

Cm(α = 45) ≤ 0 for M > 4 (9)

mtotal(b, Lref , λnose, t/c) ≤ mmax,carrier (10)

tMECO − mf uel(Lref ) × Ispg0

T0= 0 (11)

The lift and drag are calculated from the lift and dragcoefficient:

L = 0.5ρv2Sref CL(α, M, b, Lref , λnose, t/c) (12)

D = 0.5ρv2Sref CD(α, M, b, Lref , λnose, t/c) (13)

It is also assumed the fuel mass vary linearly with thespaceplane length because, since a longer vehicle meanslonger fuel tanks. The weight evolution with respect to thegeometrical changes is captured by an empirical method forweight estimation. This method is fed with the design vec-tor as well as a selected static load which will be explainedfurther. The aerodynamics of the vehicle is captured by theuse of a surrogate model.


2.4 Optimization architecture

In order to solve the problem the Non-Linear-Programmingof the trajectory optimization characteristics were used.This takes advantage from the fact that the trajectory opti-mizer already solves thousands of constraints. By addingthe geometrical design variables as state variables in thetrajectory problem and by setting their derivatives to zerothe optimizer was able to take into account all the designvariables at once. This approach is simple, intuitive and par-ticularly applicable to this problem. It gave good resultswith good convergence properties. One drawback is that itadds one constrain per discretization point per geometricaldesign variable (derivative equals to zero). This might makethe problem unnecessary larger and this must be seriously

considered for higher dimension problems. The procedureis shown in Fig. 3.

3 Optimal control problem

In order to solve the trajectory optimization problem thework of Fahroo and Ross (2002) is followed. They proposeda Chebyshev pseudo-spectral collocation method having theadvantages of using Legendre-Gauss-Lobatto (LGL) pointswhich are said to lead to a very high accuracy in the dis-cretization and low computational costs. This method isbased on a discretization of the control and/or state historywhich transforms the optimal control problem into a non-linear programing problem (NLP). An interpolation scheme

Fig. 3 Optimization process


is used to determine the time history of the control and thestate variables (Betts 2010). This approach is very efficientand particularly applicable to the choice of architecture forthe MDO problem. A Python code with Fortran wrappedroutines was developed to solve this problem.

The optimal control problem can be stated as follows:find the control vector u(τ) minimizing

J = M[x(τf ), τf ] +∫ τf

τ0

L[x(τ), u(τ ), τ ]dτ (14)

The dynamic of the system is formulated as equalityconstraints:

f [x(τ ), x(τ ), u(τ ), τ ] = 0 (15)

If ∂f/∂x is non-singular, this leads to the ordinary differ-ential equation of the system:

x(τ ) = f [x(τ), u(τ ), τ ] (16)

The pseudo-spectral approach transforms this probleminto a set of non-linear constraints, which leads to thefollowing NLP: find the coefficients

X = (x0, x1, ..., xN ), U = (u0, u1, ..., uN ) (17)

and the final time τf (if necessary) that minimize

JN(X,U, τf ) = M(xN, τf ) + τf − τ0

2

N∑k=0

L(xk, uk, τk)wk (18)

subject to

f [2/(τf − τ0)dk, xk, uk, τk] = 0, k = 0, 1, ..., N (19)

gl ≤ g(xk, uk, τk) ≤ gu, k = 0, 1, ..., N (20)

ψl ≤ ψ[x0, xf , (τf − τ0)] ≥ ψu (21)

The LGL points and the optimal weights wi are given bythe Clenshaw-Curtis scheme (Clenshaw and Alan 1960). TheLGL points are required to lie within the [−1, 1] interval,therefore the following time transformation is required witht ∈ [−1, 1]:

τ(t) = (τf − τ0)t + (τf + τ0)

2(22)

This creates a large set of constraints that has to be fed tothe optimizer.

3.1 Jacobian sparsity

It is possible to transform a large problem with a large num-ber of collocation points into a set of smaller problems withless points. This leads to more precise and faster optimiza-tion. For this purpose a knotting method (Ross and Fahroo

2004) was implemented. Assuming a knot to be placedbetween two phases with �x1 and �x2 being the states and t1iand t2i the i-th being the collocation points (which are thetime in most of the problem) of the phase one and the phase

two respectively. The system has K state variables and themass is the K-th plus one.

gk=1,...,K−1 = x1k=1,...,K−1(t

1f ) − x1

k=1,...,K−1(t20 ) = 0 (23)

In order to account for the discontinuity the followingequation is required:

gK = x1K(t1f ) − x1

K(t20 ) = 0 (24)

Having two different phases is the same as havingtwo separate smaller problems with supplementary con-straints for the consistency of the problem. Under thosecircumstances the Jacobian of the constraints has more zero-element than the unsplit problem. When the optimizer isable to recognize the zero-terms of the Jacobian, the evalu-ation of this latter becomes much faster. Figure 4 shows atypical problem without and with a knot placed in the mid-dle of the trajectory. From this figure one can observe thatthe sparsity of the Jacobian substantially increases by usinga knot. The advantage lies in the fact that as the numberof collocation points increases, the sparsity rises. The largerthe number of points the more accurate the results. Becausethe sparsity increases, the problem, although much larger interms of number of points, is not significantly more difficultto be solved by the optimizer.

Fig. 4 Jacobian sparsity without (top) and with (bottom) a knot placedin the middle of the trajectory


3.2 Adaptive collocation points mesh

The previously demonstrated advantages of splitting theproblem into several phases led to the implementation of anadaptive collocation points mesh in the code. The problemis solved at the first step on an initial mesh. Following anexisting work (Darby et al. 2011), a residual matrix R of thisinitial solution is build up in the following way:

R =∣∣∣∣DX − ti − ti−1

2F(X, U , τ )

∣∣∣∣ (25)

Where D and X are respectively the differentiation matrixand the state values at the collocation points. t is defined as

ti = ti + ti+1

2(26)

With i = 1, ..., Nc − 1, Nc being the number of colloca-tion points. The state values are interpolated on these pointsusing a Lagrange interpolation method.

The column of the obtained matrix leads to a metricallowing for iteratively adapting the mesh:

1. If the metric values of q phase have the samemagnitude,then the number of collocation points is increased inthat phase.

2. If the metric values present a peak value, then a knot isplaced at this peak separating the phase into two newphases.

This mesh adaptation proved itself to ease the optimizationand to lead to smoother results for the optimal control.

The optimizer used to solve the NLP must be able todeal with a large (thousands) number of constraints. For thisreason SNOPT (Gill et al. 2002) was chosen. This optimizerwill be the driver of the MDO problem solution. Becausethe number of constraints to evaluate is high they must bequickly evaluated in order to keep the computational timepractical. Consequently the estimation of the weight and theevaluation of the aerodynamic must be rapidly carried out.

4 Spaceplane weight estimation

The weight of the spaceplane has a very large effect on itsperformance and on the subsequent trajectory optimizationprocedure. The spaceplane follows a sub-orbital trajectory.Consequently the energy to dissipate is substantially smallerthan an orbital reentry. Most of its energy is expressed interms of potential energy instead of kinetic energy. Thechance of skipping-back into space is zero. The problemdue to heating is less severe but the load factor can beproblematic.

The reentry aerodynamic loads are used as the sizing loadcase. Indeed a possible failure in the spaceplane mission isthe mis-ejection of the US. In this case, jettisoning the US

is not possible as it carries the most valuable item of theflight: the payload. In this case, the spaceplane reenters theatmosphere with its dry weight plus the weight of the USfully wet.

High fidelity methods such as FEM do not provide read-ily the overall weight of the splaceplane. An empiricalmethod was selected for the weight estimation. Such amethod had successful results in previous similar studies(Yokoyama et al. 2007; Rowell and et al 1999).

The method selected for the weight estimation is basedon the hypersonic aerospace sizing analysis (HASA) (Harloffand Berkowitz 1988). This method uses elements from Glatt(1974), a empirical weight estimation method for advancedtransportation vehicle developed by NASA. HESA usesa series of empirical equations to estimate the weight ofdifferent weight groups of an spaceplane. These weightgroups are the structure, the propulsive stack, the hydraulicsand electrical systems, the avionics, the landing gear andthe thermal protection system. For each of these groups aweight equation is derived from statistical regressions overhypersonic vehicle data. The method has been tested againstreal vehicle values (Space Shuttle) in Harloff and Berkowitz(1988) with very good results.

Because the studied vehicle has no horizontal stabilizerand a vertical stabilization assured by two winglets, themethod is slightly modified by averaging the contributionof the vertical and horizontal stabilizers weighted respec-tively by the cosine and sine of the wingtip angles. Thismethod proved itself to be quite efficient. The results it gavewere very close to the real mass estimate of the currentlystudied vehicle that came from more classical design meth-ods. The latter statement settled the decision to use it forthe global multidisciplinary optimization. The iterative pro-cess only takes milli-seconds and was therefore carried outonline during the optimization.

5 Aerodynamic analysis

The Stanford’s University Unstructured Code (SU2) (Pala-cios et al. ) was used to carry out the aerodynamic analysis. Itis a multi-physics code with a wide range of capabilities andparallel computation support. It possesses an optimizationready framework, which makes it very suitable for aircraftoptimization.

In this study, an Euler formulation was used. Conse-quently the viscous terms of the Navier-Stokes equationswere not considered. The choice for such a simulation canbe justified by the following reasons:

1. Most of the flight regimes are at very high Mach num-bers. At these speeds, the viscous effects are largelyovercome by the wave drag. Consequently taking into


account the viscous terms has less effect than at lowerairspeeds.

2. The discretization of the Euler equations creates asecond-derivative term from the Taylor expansion. Thisterm creates an artificial dissipation that partially simu-lates the viscous drag.

3. The mesh deformation procedure for large geometrydeformations has a destructive effect on the viscouslayer of the mesh. Therefore a viscous mesh could notbe employed and fluid analysis such as RANS could notbe performed.

The CAD model of the spaceplane was used to generatean unstructured, non-viscous mesh using Pointwise. Figure5 shows an example of the results obtained with SU2 forsolving the Euler equations for a typical Mach number andangle of attack. In this figure the spaceplane flies at Mach 4and 25 degrees of angle of attack.

The gradients were obtained using the adjoint formu-lation (Jameson 1988; Pironneau 1994) implemented in SU2.Accurate gradients of the aerodynamic coefficients withrespect to the Mach number, α and geometrical defor-mations were calculated using this method. The gradientsare used to provide information for the Gaussian processgradient enhancement.

The geometry deformations were applied directly tothe computational mesh using the free-form deformation(FFD)(Sederberg and Parry 1986) tool available in SU2.Samareh (Samareh 2004) demonstrated the advantages ofthese techniques: they avoid to place a CAD software inthe mesh deformation loop which greatly eases the pro-

Fig. 5 SU2 calculation results at Mach 4 and 25 degrees of angle ofattack

cess and they also reduce drastically the number of designvariables as the mesh nodes positions are parametrized bya limited number of control points. Instead of deformingdirectly the surface of the object, the FFD creates a defor-mation field over the entire domain. Therefore the FFDmethod deforms the object independently of its geometricaldescription.

During the mesh deformation using FFD some possibletangling of the mesh was observed, particularly when largedeformations were applied. To overcome this problem, amesh repairing technique was used (Escobar et al. 2005). Anoptimization procedure is set up using a quality measure ofthe mesh to repair any possible tangling of the mesh. Thismethod was used after every geometry deformation in orderto ensure that the mesh fed to the CFD solver does not havenegative volume cells.

One aerodynamic evaluation, including the sensitivitiescalculation, takes approximately two minutes on a 700-corecluster to converge (the convergence of the solution wasdefined using a Cauchy criterion of ε = 10−5). Trajectoryoptimization using the pseudo-spectral method as explainedearlier requires a large number of aerodynamic analysis (103

order of magnitude). This high number of evaluations is aconsequence of the pseudo-spectral method. It requires thevalidation of thousands of constraints as the equation ofmotions are expressed as a set of equality constraints onevery collocation point. These constraints and their Jacobianmust be evaluated for every optimizer iteration. The com-putational burden can become significant if the evaluationmethod is computationally demanding. To overcome thisproblem a series of response surfaces of the aerodynamiccoefficients are created using a Gaussian process method-ology where a new approach for a non-stationary kernel isproposed.

6 Gaussian process regression for response surfacecreation

The need for a prompt evaluation of the aerodynamic coeffi-cient is answered by the creation of response surfaces. Thisapproach is commonly found in MDO problems and vari-ous techniques were investigated as in the work of Colonno(2007), Yokoyama et al. (2007) and Kaletta et al. (2004).

In this study a Gaussian process regression (GPR)approach is selected due to its ability to fit a dimensionallysparse dataset and to provide a covariance of the estimatedenabling a reading of the fit quality. Due to the avail-ability of the gradients of the aerodynamic coefficients,thanks to the adjoint formulations, the dataset information isenriched with the derivatives information. The readiness ofthe derivatives is also a motivation to take a novel approachto the creation of a non-stationary kernel.


6.1 Dataset sampling

In this study one of the most common methods for datasampling was chosen: the Latin Hyper Cube sampling (orLHC sampling), which was first proposed by McKay et al.(1979). It is a popular Design of Experiment (DoE) meth-ods for GPRs. However in the present study the physicsof the problem gives additional information. The aerody-namic coefficients of the vehicle displayed a highly varyingbehavior between Mach 0 and 2 and relatively constantbehavior for higher Mach numbers. Six design variables areused to generate the samples: Mach number, angle of attackand four geometrical variables b, Lref , λnose and t/c. Thedesign domain was split into two sub-domains, in whichdifferent LHC sampling with different densities were per-formed. This leads to a biased spatial uniformity of theDoE in the dimension of the Mach number with the advan-tage that the variations occurring during the transonic andlow supersonic phases have better chances to be properlycaptured.

6.2 Gradient enhanced gaussian process

Using the obtained dataset x from the CFD calculations, theGPR, as described by Rasmussen (2006), allows for inter-polating any desired point x∗. Describing the dataset as amultivariate Gaussian and the point of interest as anotherGaussian, it is possible to use the Bayesian inference ruleto obtain the following estimation distribution, with y∗ thedesired value, y the dataset andKij are the covariance matrixbetween the points i and j .

p(y∗/y) ≡ N (Kx∗xK−1xx y,Kxx − Kx∗xK−1

xx KTx∗x) (27)

The expected value of y∗, y∗ and its variance squared σ 2y∗

are:

y∗ = Kx∗xK−1xx y (28)

σ 2y∗ = Kxx − Kx∗xK−1

xx KTx∗x (29)

Where Kxx and Kx∗x are the covariance matrices, or ker-nels, respectively between the dataset and itself and theinterpolated point and the dataset.

GPR can be greatly improved, especially when the num-ber of dimensions is large, by adding the gradient infor-mation to the Gaussian object. The additional informationallows for reaching a satisfactory fit with less data points.This technique is sometimes referred to as co-Kriging orgradient enhanced Kriging as the work of Dwight andHan (2009). Because most of the disciplinary analysis in

aircraft design can usually provide gradients, these meth-ods are particularly relevant when the gradient informa-tion is added to the function value inside the regressionprocess.

As proposed by Solak et al. (2003) the kernel is extendedwith the derivatives of the exponential covariance function:

k(∂p

∂xi

, q) = ∂k(p, q)

∂xi

∣∣∣∣q

(30)

k(p,∂q

∂xj

) = ∂k(p, q)

∂xj

∣∣∣∣p

(31)

k(∂p

∂xi

,∂q

∂xj

) = ∂

∂xi

(∂k(p, q)

∂xj

∣∣∣∣q

)∣∣∣∣∣p

(32)

In order to add the gradient informations to the functionvalue it is assumed that the function and its gradients arecorrelated. The covariance of the initial distribution is thenaugmented with the covariance between the function and itsgradients forming the following composite kernel:

K =[

k(p, q) k(∂p∂xi

, q)

k(p,∂q∂xj

) k(∂p∂xi

,∂q∂xj

)

](33)

The dataset is also augmented with the gradients ∇y. Asmentioned before, the gradients of the aerodynamic coef-ficients are obtained using an adjoint formulation. Thismethod allows for a rapid evaluation of the derivatives of theaerodynamic functions with respect to the angle of attack,the Mach number and the geometrical parameters:

Y =[

y

∇y

](34)

Fig. 6 Pitching moment coefficient of the spaceplane estimation usingGPR with a stationary Kernel as a function of the Mach number forAoA of 15 degree. The red dots corresponds to the data sample. Theblue line is the GPR estimate


Adding the gradients information greatly helps to reducethe number of points needed for an acceptable fit. Herelies a strong advantage of using the adjoint for the gradi-ent estimations. Because the derivatives in every dimensionscan be obtained in only one calculation, the computationaltime required to obtain an accurate surrogate model canbe greatly reduced as opposed to simply refining the datasample by adding training points. However as identifiedby Dwight and Han (2009) the gradient information candeteriorate the quality of the fit if the derivatives are notaccurate enough. A very promising solution proposed byLukaczyk et al. (2013) is to relax the ’shoot-angle’ of thegradient by adding noise. In other words the earlier assump-tion of an exact correlation between the gradient and theprior is relaxed. A new covariance matrix is used for theGP:

K = K + Kε (35)

Where Kε is a noise component added to the initial covari-ance distribution. A useful noise model is an independent

Fig. 7 Pitch moment coefficient GPR and corresponding LGP asfunction of the Mach number for AoA 15 degree. Non-StationaryKernel GPR

Gaussian noise with null mean (Lukaczyk et al. 2013). Thisresults in:

Kε =[

σ 2n,y 0

0 σ 2n,∇y

](36)

where σn,y, σn,∇y are hyperparameters to be selected. Thefact of having two different noises for the objective andthe gradient relaxes the assumption on their correlation

Stationary Kernel GPR

Proposed Non-Stationary Kernel GPR

Fig. 8 Validation of the new GPR method and comparison with astationary GPR using an axial force coefficient dataset


Fig. 9 Aerodynamic ResponseSurface generation flowchart

and greatly improves the fit quality in case of inaccurategradients.

6.3 Stationary Kernel and Hyperparameters selection

A common choice to create the kernels in GPR is to use asquared-exponential function. The covariance between thepoint p and q is given by:

k(p, q) = σ 2f exp

[− 1

2θ2

N∑k=1

(pk − qk)2

](37)

where N is the spatial dimension, and pk and qk are the k-thcomponents of the p and q vectors respectively. σf and θ arethe parameters of the covariance function and consequentlythe hyperparameters of the GPR.

The choice of the hyperparameters has a very large influ-ence on the quality of the fit. To select the appropriate

hyperparameters, it is proposed to minimize the followingmarginal likelihood function (Rasmussen 2006):

J = −1

2YT [K]−1Y − 1

2log ‖[K]‖ − n

2log 2π, (38)

This results in an optimization problem:

minσf ,θ,σn,y ,σn,∇y

J, (39)

This optimization problem is solved using the CovarianceMatrix Adaptation Evolution Strategy or CMA-ES (Hansen

2006). This method is a global optimization, which is ben-eficial since it is difficult to give an initial solution for thehyperparameters before-hand.

The selection of the hyperparameters enlightens oneof the main problems encountered when using GPR foraerospace applications: because the hyperparameters arechosen globally, they cannot be adapted for both a sharply


Fig. 10 Drag coefficient response surface obtained with SU2

varying region (the transonic region) and a flat region (thesupersonic region). These type of results, which exhibit vari-able smoothness and non-linearities, are very frequent inthe field of aerodynamics because of the transonic effects.In our case, the trajectory is significantly impacted by thesound barrier passage. Therefore it is important to capturethe Mach one region properly. An extreme example is givenin Fig. 6, which shows the evolution of the spaceplane pitch-ing moment coefficient as a function of the Mach numberfor a 15 degree angle of attack. In this figure it is clear thatthe GPR cannot find the parameters that fit the sharply vary-ing coefficient from Mach 0 to Mach 4 and the plateau forhigher speeds. Therefore in order to properly capture theaerodynamic of the spaceplane a non-stationary kernel isrequired.

Fig. 11 Lift coefficient response surface obtained with SU2

Fig. 12 Optimal Control Problem solution

6.4 Non-stationary kernel

An important requirement for a non-stationary kernel is thecontinuity of the second derivative of the resulting GPR.The trajectory optimization can become ill-behaved if thiscondition is not respected and the whole MDO can fail(Betts 2010). Consequently splitting the domain and havinglocally adapted kernels for each sub-domain is not desir-able (Gramacy and Lee 2008). Instead a smoothly varying θ isrequired. In this study a new technique is developed inspiredby the work of Plagemann et al. (Lang et al. 2007), that takesadvantages of the readily available gradients.

In order to develop a non-stationary kernel, the non-stationary covariance function described by Paciorek andSchervish (2004) is used as the starting point:

k(p, q) = σf |�p| 14 |�p| 14 |�q | 14∣∣∣∣�p + �q

2

∣∣∣∣− 1

2

× exp

[−(p − q)T

(�p + �q

2

)−1

(p − q)

](40)

The Local Adaptation Kernels (LAKs) �p and �q transmitthe local variation of the hyperparameters with respect tothe spatial variations between the points p and q to the GPcovariance matrix K.

Table 1 Optimized parameters and performance index gain

Parameters Optimized value ratio

b∗ 1.14719

l∗ 1.0324

λ∗ 0.9047

t/c∗ 0.942954

M∗injected 1.0749


Fig. 13 Optimized and baseline suborbital spaceplane shape

In order to obtain the LAKs the benefit of having accessto the gradient information is used. When the variation ofthe process is high, the equivalent θ must be small. Whenthe variation of the process is small, the equivalent θ must behigh. This intuition leads to the following definition of �:

� = �

((∣∣∣∣∂y∂x∣∣∣∣)−1

)(41)

With ∂y∂x =

[∂y∂x1

, ...,∂y

∂xN

], andN is the number of dimensions.

A simple yet efficient formulation is defined:

� =[(∣∣∣∣ ∂y

∂x

∣∣∣∣∣∣∣∣)−1

]·[(∣∣∣∣ ∂y

∂x

∣∣∣∣∣∣∣∣)−1

]T

(42)

This leads to an N × N diagonal � matrix. Note that theinverse operator acts element-wise here and | · | is theelement-wise absolute value operator.

The LAKs for the prior dataset are readily availablebecause the derivatives exist at the data points. Howeverwhen a new point must be interpolated the derivatives arenot available. For this reason Latent Gaussian Processes

Fig. 14 Pitch moment of the optimized vehicle above Mach 4 alongwith the trim line

(LGPs) are used. For every dimension, a GP is created basedon the derivatives in this dimension on the global designspace. This allows for interpolating the derivatives at anypoint in the design space and sending up this information tothe LAKs at this point. LAKs are then available at any point.It is necessary to create a LAKs for every dimension becausethe local variation can occur in only one dimension. Forexample the drag coefficient presents such behavior onlyalong the Mach number whereas along the angle of attackthe behavior is smoothly quadratic.

The LAKs are defined at any point p of the design spaceas:

�p = [ξp

] · [ξp

]T (43)

Where ξp = [ξp,1, ..., ξp,N

]is defined using a conventional

GPR described by (28):

ξp,i = K∗p,xK∗−1

x,x ωi (44)

K∗ is the covariance matrix formed with the stationaryexponential covariance function described in (37). ωi , i =1, ..., N is the metric constructed with the i-th derivative atthe m-point xs of the dataset defined as:

ωi =[(∣∣∣∣ ∂y1

∂x1,i

∣∣∣∣)−1

,

(∣∣∣∣ ∂y2

∂x2,i

∣∣∣∣)−1

, ...,

(∣∣∣∣ ∂ym

∂xm,i

∣∣∣∣)−1

](45)

In order to avoid ill-posed LGPs, a bandwidth factor isused to damp strong variation in the derivative dataset. Thispermits to avoid negative values in the LAKs and to reduceundesirable oscillations in the LGP. The k-th metric of thedataset in the i-th dimension ωk,i is damped with the factora as follow:

ωk,i = min(ωk,i , a · min(ωi)) (46)

The factor a is added to the hyperparameters to be opti-mized. The hyperparameters of the LGPs also have to beoptimized. They are exponential covariance functions (3hyperparameters) as well as a noise parameter. In this study,only one dispersion parameter and noise parameter wereoptimized for all the LGPs whereas one length scale perLGP was added to the set of the hyperparameters. Thishelps to reduce the exponentially increasing complexity ofthe optimization process with the number of dimensions. Aspointed by Ying and et al (2007) an increasing number ofhyperparameters to optimize can lead to a difficult optimiza-tion problem. With this approach this number is kept lowand the optimization of the hyperparameters can be solvedin approximatively 5 minutes on a standard workstation.A new marginal likelihood function is defined, inspired byPlagemann et al. (2008):

JNS = −1

2YT [K]−1Y − 1

2log ‖[K]‖

−N∑

i=1

(1

2log ‖[K∗

i ]‖)

− n

2log 2π (47)


Fig. 15 Optimized trajectory radius and longitude as functions of the time

Where K∗i is the i-th LGP covariance matrix.

This defines a new optimization problem:

minσf ,θ,σn,y ,σn,∇y ,σl ,σn,l ,θl,1,...,θl,N

JNS (48)

This proposed method led to very appreciable results. Forinstance the previously described pitching moment coef-ficient was very difficult to be fitted properly with thestationary approach. Figure 7 shows the improvement on thefit using the Latent GPs non-stationary approach.

The accuracy of the new regression method is investi-gated using an ordered dataset of the axial force coefficientCX. The ordered dataset is compared to the values obtainedusing the GPR trained with 25 points sampled with a LHCmethod. The comparison as well as the covariance of theestimates is shown in Fig. 8 along with the regression of thesame dataset using a stationary kernel in order to show theperformances of the proposed method. None of the pointcompared in the figure was part of the training dataset of

the GPR. Figure 8 shows clearly the better fit capability ofthe non-stationary GPR. This validation showed an averageerror of 1.19 % for the newly developed non-stationary GPRagainst 12.67 % for the stationary GPR.

6.5 Aerodynamic response surfaces

The aerodynamic calculations were used to iteratively createthe response surfaces using the process pictured in Fig. 9.This process is repeated until the fit quality converges.

Figures 10 and 11 respectively show the drag and thelift response surfaces created using the results obtained withSU2. The variable smoothness and non-linearities of thesephenomenon, a problem for the GPs as mentioned earlier,is clear on these figures. Such response surfaces are createdfor the aerodynamic coefficient with respect to the designvector, i.e. the angle of attack, the Mach number and thegeometrical parameters. They are used to promptly obtainedthe aerodynamic coefficients during the MDO process.

Fig. 16 Optimized trajectory latitude and velocity as functions of the time


Fig. 17 Optimized trajectory flight path angle and azimuth as functions of the time

7 Optimization results

With the help of the previously described methods theMDO problem was successfully solved. The optimal controlsolution is given in Fig. 12.

The optimized parameters are given in Table 1 in the formof the ratio between the optimal and the baseline. The opti-mized shape is displayed in Fig. 13 on top of the baselineshape. The red one is the optimized shape and the blue onethe baseline.

The pitching moment of the optimized vehicle is shownin Fig. 14 along with the trim line (Cm = 0). It is clear onthis figure that over Mach 4 the vehicle is trimmed betweenα = 35 and α = 45 and is longitudinally statically stable(Cm,α < 0).

The results of this optimization led to the trajectorypresented in Figs. 15, 16 and 17.

Because of those constraints that are linked to the mis-sion, mainly the separation from the carrier issue, the opti-mizer did not tend to change the spaceplane into a purerocket. Despite the mass and drag gained by adding wingsthis can be compensated by taking advantage of the liftduring the ascent. From the baseline, the optimal shape islonger. This gives a larger quantity of fuel and the space-plane is consequently able to have engine thrust for a longertime. This comes with a drag and a mass penalty. Howeverthe drag becomes really small in a short time as the space-plane ascent due to the exponentially decreasing air density.The span is broader, the main positive effect is a gain in L/D

ratio but comes with a mass penalty. Having a higher glideratio means that the spaceplane is able to have a better lift-assisted climb. It is therefore able to reach quickly a verylow density atmosphere and consequently with very lowdrag. This might compensate the drag penalty obtained withthe increase in length. The thinner wing, although heav-ier, also compensates for the drag penalty. It seems that the

optimizer tended to increase as much as possible the aero-dynamic efficiency in order to compensate the drag gainedby lengthening the spaceplane. Perhaps this compensationstopped when the mass penalty because of increasing theaerodynamic efficiency became too high and jeopardizedthe process. The increase in the double delta has almostno effect on the aerodynamic efficiency and has very littleeffect on the mass. It seemed natural to say that its modifi-cation was due to the constrain on the longitudinal stabilityin the hypersonic regime and that having more double deltawas necessary to fulfill this constrain.

8 Conclusion

This study demonstrated the optimization of a sub-orbitalspaceplane using a coupling between a geometrical opti-mization, mainly based on aerodynamic and structuraloptimization, with an optimal control problem. The aero-dynamics was calculated using an Euler code and usedFree-Form Deformation in order to apply the geometricalchanges directly to the mesh in a safe manner using a meshuntangling procedure. The sensitivities of the aerodynamicobjectives, lift, drag and pitching moment were obtainedusing an adjoint formulation of the Euler equations forthe aforementioned FFD transformations. The results werethen used to feed a Gradient Enhanced Gaussian ProcessRegression algorithm for which a non-stationary Kernelmethod based on local variation estimates using latent pro-cesses was specifically developed and implemented in aPython-Fortran combination. This non-stationary approachhas the advantage of being able to fit properly a vari-able smoothness prior such the aerodynamic of the vehi-cle over its whole Mach range. The weight estimationwas based on empirical methods developed by a NASAcontractor and the program Hypersonic Aerospace Sizing


Analysis (HASA) was used. The trajectory optimal con-trol problem was solved using a Chebyshev Pseudo-Spectralmethod with collocation points mesh refinement and adap-tation. For the latter analysis a dedicated program waswritten as a combination, once more, of Python and For-tran. The sequential quadratic programming optimizer usedwas SNOPT which is very suited choices for solvingNLPs.

The multidisciplinary optimization problem was definedas an integration of the geometry optimization and theoptimal control formulation. The optimizer of the opti-mal control problem was used to optimize both the geo-metrical and control parameters at the same time. Theobtained results were satisfactory. The geometrical param-eters were modified less than a maximum of 15 %. Themain reason for the small modification in the geometri-cal parameters is the fact the baseline shape originatesfrom a pre-existing design, the european space shuttleproject Hermes. The optimal trajectory coupled with theoptimal shape led to a 7.49 % increase in the injectedmass.

Acknowledgments This study was conducted within Swiss SpaceSystems facilities and all the company resources were made available.The authors would like to gratefully thank the S3 staff for their helpand support.

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricteduse, distribution, and reproduction in any medium, provided you giveappropriate credit to the original author(s) and the source, provide alink to the Creative Commons license, and indicate if changes weremade.

References

Betts JT (2010) Practical methods for optimal control and estimationusing nonlinear programming, vol 19. SIAM

Clenshaw CW, Alan RC (1960) A method for numerical integration onan automatic computer. Numerische Mathematik 2.1, pp 197–205

Colonno MR (2007) A multidisciplinary optimization methodologyfor rocket vehicle systems

Darby CL, Hager WW, Rao AV (2011) An adaptive pseudospectralmethod for solving optimal control problems. Opt Control ApplMethods 32(4):476–502

Dwight RP, Han Z-H (2009) Efficient uncertainty quantifi-cation using gradient-enhanced kriging. 11th AIAA Non-Deterministic Approaches Conference. AIAA Paper, VA, USA,pp 2009–2276

Escobar JM, Montenegro R, Montero G, Rodriguez E, Gonzalez-Yuste JM (2005) Smoothing and local refinement tech-niques for improving tetrahedral mesh quality. Comput Struct83(28)

Fahroo F, Ross IM (2002) Direct trajectory optimization by a Cheby-shev pseudospectral method. Journal of Guidance Control, andDynamics 25.1, pp 160–166

Gill PE, Murray W, Michael AS (2002) SNOPT: An SQP algo-rithm for large-scale constrained optimization. SIAM journal onoptimization 12.4, pp 979–1006

Glatt CR (1974) WAATS: A Computer Program for WeightsAnalysis of Advanced Transportation Systems. NASACR-2420

Gramacy RB, Lee HKH (2008) Bayesian treed Gaussian process mod-els with an application to computer modeling. J Am Stat Assoc103:483

Hansen N (2006) The CMAEvolution Strategy: A Comparing Review.In: Lozano JA, Larraga P, Inza I, Bengoetxea E (eds) Towards anew evolutionary computation. Advances in estimation of distri-bution algorithms. Springer, pp 75–102

Harloff GJ, Berkowitz BM (1988) HASA, Hypersonic Aerospace Siz-ing Analysis, for the Preliminary Design of Aerospace Vehicles.National Aeronautics and Space Administration

Jameson A (1988) Aerodynamic design via control theory. SpringerKaletta P, Wolf K, Fischer A (2004) Structural Optimization in

Aircraft Engineering using Support Vector Machines. In: Opera-tions research proceedings 2003 (pp. 411-418). Springer, BerlinHeidelberg

Lang T, Plagemann C, Burgard W (2007) Adaptive Non-StationaryKernel Regression for Terrain Modeling. Robotics: Science andSystems

Lukaczyk T, Palacios F, Alonso JJ (2013) Managing Gradient Inac-curacies while Enhancing Optimal Shape Design Methods. 51stAIAA Aerospace Sciences Meeting

Martins RA, Joaquim R, Alonso JJ, Reuther JJ (2004) High-fidelityaerostructural design optimization of a supersonic business jet.Journal of Aircraft 41.3:523–530

McKay MD, Beckman RJ, Conover WJ (1979) Comparisonof three methods for selecting values of input variablesin the analysis of output from a computer code, vol 21,pp 239–245

Paciorek C, Schervish M (2004) Nonstationary covariance func-tions for Gaussian process regression. In: Proceedings ofthe Conference on Neural Information Processing Systems(NIPS)

Palacios F, Colonno MR, Aranake AC, Campos A, CopelandSR, Economon TD, Lonkar AK, Lukaczyk TW, Tay-lor TWR, Alonso JJ Stanford University Unstructured(SU2): An open-source integrated computational environ-ment for multi-physics simulation and design, AIAA Paper2013-0287

Pironneau O (1994) Optimal shape design for aerodynamics. AGARDREPORT 803

Plagemann C, Kersting K, Wolfram B (2008) Nonstationary Gaus-sian process regression using point estimates of local smoothness.Machine learning and knowledge discovery in databases. Springer,Berlin Heidelberg, pp 204–219

Rasmussen CE (2006) Gaussian processes for machine learning. Cite-seer

Reuther J, Jameson A (1995) Supersonic wing and wing-body shapeoptimization using an adjoint formulation. Research Institute forAdvanced Computer Science, NASA Ames Research Center

Rowell LF, Braun RD, Olds JR, Unal R (1999) Multidisciplinaryconceptual design optimization of space transportation systems.Journal of Aircraft 36.1, pp 218–226

Rowell LF et al (1999) Multidisciplinary conceptual design optimiza-tion of space transportation systems. Journal of Aircraft 36.1,pp 218–226

Samareh JA (2004) Aerodynamic shape optimization based on free-form deformation. AIAA paper, 4630:2004

Sederberg TW, Parry SR (1986) Free-form deformation of solid geo-metric models. In: ACM Siggraph Computer Graphics, vol 20.ACM, pp 151–160

http://creativecommons.org/licenses/by/4.0/

http://creativecommons.org/licenses/by/4.0/


Sobieszczanski-Sobieski J, Haftka RT (1997) Multidisciplinaryaerospace design optimization: survey of recent developments.Struct Opt 14(1):1–23

Solak E, Murray-Smith R, Leithead W, Rasmussen C, LeithD (2003) Derivative observations in Gaussian Process mod-els of dynamic systems. Adv Neural Inf Process Syst 15:1057–1064

Ross IM, Fahroo F (2004) Pseudospectral knotting methods for solv-ing nonsmooth optimal control problems. Journal of Guidance,Control, and Dynamics 27.3, pp 397–405

Takeshi T, Mori T (2004) Optimal conceptual design of two-stage reusable rocket vehicles including trajectory optimiza-

tion. Journal of Spacecraft and Rockets 41.5, pp 770–778

Tsuchiya T, Takenaka Y, Taguchi H (2007) Multidisciplinary designoptimization for hypersonic experimental vehicle. AIAA journal45.7, pp 1655–1662

Ying X et al (2007) A non-stationary covariance-based Kriging methodfor metamodelling in engineering design. International Journal forNumerical Methods in Engineering 71.6, pp 733–756

Yokoyama N, Suzuki S, Tsuchiya T, Taguchi H, Kanda T (2007)Multidisciplinary design optimization of space plane consideringrigid body characteristics. Journal of Spacecraft and Rockets 44.1,pp 121–131

Documents

Trajectory driven multidisciplinary design optimization of